an:07217853
Zbl 1451.14129
Ohno, Masahiro
Nef vector bundles on a projective space with first Chern class three
EN
Rend. Circ. Mat. Palermo (2) 69, No. 2, 425-458 (2020).
00450512
2020
j
14J60 14F06
nef vector bundles; Fano bundles; spectral sequences
Given a nef vector bundle \(\mathcal{E}\) on a projective space \(\mathbb{P}^n\) it is well-known that \(c_1(\mathcal{E})\geq 0\). Nef vector bundles \(\mathcal{E}\) with \(c_1(\mathcal{E})\leq 2\) were classified
by [\textit{T. Peternell} et al., Lect. Notes Math. 1507, 145--156 (1992; Zbl 0781.14006)] analyzing the contraction morphisms of extremal rays. In particular, for \(n\geq 2\), \(\mathbb{P}(\mathcal{E})\) is a Fano variety. A different proof of the classification was obtained by \textit{M. Ohno} [``Nef vector bundles on a projective space or a hyperquadric with the first Chern class small'', Preprint, \url{arXiv:1409.4191}] using the twists \(\mathcal{E}(d)\).
The paper under review deals with the next case, namely nef vector bundles \(\mathcal{E}\) on \(\mathbb{P}^n\) (over an algebraically closed field of characteristic zero) with \(c_1(\mathcal{E})=3\) are completely classified. In particular, one has \(0\leq c_2(\mathcal{E})\leq c_1(\mathcal{E})^2=9\). When \(c_2(\mathcal{E})<8\), the author proves that the nef vector bundles \(\mathcal{E}\) are globally generated. For \(c_2=8\) and \(9\), there exist examples of non-globally generated nef vector bundles on the projective plane.
Joan Pons-Llopis (Ma??)
Zbl 0781.14006